title: Eigenstates and instabilities of chains with embedded defects.
abstract: We consider the eigenvalue problem for one-dimensional linear Schrödinger lattices (tight-binding) with an embedded few-sites linear or nonlinear, Hamiltonian or non-conservative defect (an oligomer). Such a problem arises when considering scattering states in the presence of (generally complex) impurities as well as in the stability analysis of nonlinear waves. We describe a general approach based on a matching of solutions of the linear portions of the lattice at the location of the oligomer defect. As specific examples, we discuss both linear and nonlinear, Hamiltonian and PT-symmetric dimers and trimers. In the linear case, this approach provides us a handle for semi-analytically computing the spectrum [this amounts to the solution of a polynomial equation]. In the nonlinear case, it enables the computation of the linearization spectrum around the stationary solutions. The calculations showcase the oscillatory instabilities that strongly nonlinear states typically manifest.
authors.affiliation: Department of Mathematics, Bard College, Annandale-on-Hudson, New York 12504, USA.
authors.forename: J
authors.initials: J
authors.lastname: D'Ambroise
authors.forename: P G
authors.initials: PG
authors.lastname: Kevrekidis
authors.forename: S
authors.initials: S
authors.lastname: Lepri